Nontrivial t-designs over finite fields exist for all t

A t - ( n , k , λ ) design over F q is a collection of k-dimensional subspaces of F q n , called blocks, such that each t-dimensional subspace of F q n is contained in exactly λ blocks. Such t-designs over F q are the q-analogs of conventional combinatorial designs. Nontrivial t - ( n , k , λ ) designs over F q are currently known to exist only for t ⩽ 3 . Herein, we prove that simple (meaning, without repeated blocks) nontrivial t - ( n , k , λ ) designs over F q exist for all t and q, provided that k > 12 ( t + 1 ) and n is sufficiently large. This may be regarded as a q-analog of the celebrated Teirlinck theorem for combinatorial designs.